geometry problems from Pan-American Girls' Mathematical Olympiad (PAGMO) with aops links in the names
2021
Consider the isosceles right triangle ABC with \angle BAC = 90^\circ. Let \ell be the line passing through B and the midpoint of side AC. Let \Gamma be the circumference with diameter AB. The line \ell and the circumference \Gamma meet at point P, different from B. Show that the circumference passing through A,\ C and P is tangent to line BC at C.
Let ABC be a triangle with incenter I, and A-excenter \Gamma. Let A_1,B_1,C_1 be the points of tangency of \Gamma with BC,AC and AB, respectively. Suppose IA_1, IB_1 and IC_1 intersect \Gamma for the second time at points A_2,B_2,C_2, respectively. M is the midpoint of segment AA_1. If the intersection of A_1B_1 and A_2B_2 is X, and the intersection of A_1C_1 and A_2C_2 is Y, prove that MX=MY.
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